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Metamath Proof Explorer

Theorem 4atexlemntlpq

Description: Lemma for 4atexlem7 . (Contributed by NM, 24-Nov-2012)

Ref Expression
Hypotheses 4thatlem.ph ⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) )
4thatlem0.l ⊢ ≤ = ( le ‘ 𝐾 )
4thatlem0.j ⊢ ∨ = ( join ‘ 𝐾 )
4thatlem0.m ⊢ ∧ = ( meet ‘ 𝐾 )
4thatlem0.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4thatlem0.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4thatlem0.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
4thatlem0.v ⊢ 𝑉 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 )
Assertion 4atexlemntlpq ( 𝜑 → ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) )

Proof

Step Hyp Ref Expression
1 4thatlem.ph ⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) )
2 4thatlem0.l ⊢ ≤ = ( le ‘ 𝐾 )
3 4thatlem0.j ⊢ ∨ = ( join ‘ 𝐾 )
4 4thatlem0.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 4thatlem0.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 4thatlem0.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 4thatlem0.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8 4thatlem0.v ⊢ 𝑉 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 )
9 1 2 3 4 5 6 7 8 4atexlemtlw ⊢ ( 𝜑 → 𝑇 ≤ 𝑊 )
10 1 4atexlemkc ⊢ ( 𝜑 → 𝐾 ∈ CvLat )
11 1 2 3 4 5 6 7 4atexlemu ⊢ ( 𝜑 → 𝑈 ∈ 𝐴 )
12 1 2 3 4 5 6 7 8 4atexlemv ⊢ ( 𝜑 → 𝑉 ∈ 𝐴 )
13 1 4atexlemt ⊢ ( 𝜑 → 𝑇 ∈ 𝐴 )
14 1 2 3 4 5 6 7 8 4atexlemunv ⊢ ( 𝜑 → 𝑈 ≠ 𝑉 )
15 1 4atexlemutvt ⊢ ( 𝜑 → ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) )
16 5 3 cvlsupr5 ⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ) ∧ ( 𝑈 ≠ 𝑉 ∧ ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) → 𝑇 ≠ 𝑈 )
17 10 11 12 13 14 15 16 syl132anc ⊢ ( 𝜑 → 𝑇 ≠ 𝑈 )
18 17 adantr ⊢ ( ( 𝜑 ∧ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑇 ≠ 𝑈 )
19 1 4atexlemk ⊢ ( 𝜑 → 𝐾 ∈ HL )
20 1 4atexlemw ⊢ ( 𝜑 → 𝑊 ∈ 𝐻 )
21 19 20 jca ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
22 21 adantr ⊢ ( ( 𝜑 ∧ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
23 1 4atexlempw ⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
24 23 adantr ⊢ ( ( 𝜑 ∧ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
25 1 4atexlemq ⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
26 25 adantr ⊢ ( ( 𝜑 ∧ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑄 ∈ 𝐴 )
27 13 adantr ⊢ ( ( 𝜑 ∧ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑇 ∈ 𝐴 )
28 1 4atexlempnq ⊢ ( 𝜑 → 𝑃 ≠ 𝑄 )
29 28 adantr ⊢ ( ( 𝜑 ∧ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑃 ≠ 𝑄 )
30 simpr ⊢ ( ( 𝜑 ∧ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) )
31 2 3 4 5 6 7 lhpat3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ¬ 𝑇 ≤ 𝑊 ↔ 𝑇 ≠ 𝑈 ) )
32 22 24 26 27 29 30 31 syl222anc ⊢ ( ( 𝜑 ∧ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ¬ 𝑇 ≤ 𝑊 ↔ 𝑇 ≠ 𝑈 ) )
33 18 32 mpbird ⊢ ( ( 𝜑 ∧ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) → ¬ 𝑇 ≤ 𝑊 )
34 33 ex ⊢ ( 𝜑 → ( 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) → ¬ 𝑇 ≤ 𝑊 ) )
35 9 34 mt2d ⊢ ( 𝜑 → ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) )